We analyze the relation between a topological coset model based on super SL(2, R)/U(1) coset and non-critical string theory by using free field realization. We show that the twisted N=2 algebra of the coset model can be naturally transformed into that of non-critical string. The screening operators of the coset models can be identified either with those of the minimal matters or with the cosmological constant operator. We also find that another screening operator, which is intrinsic in our approach, becomes the BRST non-trivial state of ghost number 0 (generator of the ground ring for c=1 gravity). The relation between non-critical strings and topological field theories is the subject of current interest. It has long been suggested that the latter theories describe the unbroken phase of gravity,1 but their precise relation has not been clear. It has been known that the twisting of N=2 superconformal field theory gives rise to topological theory.1,2 This suggests that any non-critical string theories may have hidden N=2 superconformal symmetry. Indeed, several authors have observed that the BRST current and the antighost field b(z) generate an algebra that is quite similar but apparently not identical to the N=2 superconformal algebra.3 It turns out that the BRST current can be modified by total derivative terms so that the antighost and the physical BRST current exactly generate a topologically twisted N=2 superconformal algebra.4,5 This does not identify, however, the structure of the models with N=2 symmetry. Recently, rather non-trivial correspondence between super SL(2, R)k/U(1) coset model6 and c=1 string has been analyzed through twisted N=2 structure. Mukhi and Vafa7 have revealed an amazing correspondence between these two models for k=3. In this letter, we discuss the relation of these models and the generalization of the correspondence to the minimal models coupled to gravity by means of the free field realization. We find that there is another interesting correspondence for k=1. Super SL(2, R)k/U(1) model is described by the bosonic coset model of SL(2, R)k×U(1)/U(1).8 For a representation of SL(2, R)k, we use the following
Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications
